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Guidelines on the calibration of non-automatic weighing instruments

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This publication is identical to the original EURAMET "Guidelines on the calibration of non-automatic weighing instruments" (EURAMET/cg-18/v.02). The copyright of the original document is held by © EURAMET e.V. 2007. The Calibration Guide may not be copied for resale and may not be reproduced other than in full. In no event shall EURAMET, the authors or anyone else involved in the creation of the document be liable for any damages whatsoever, arising out of the use of the information contained herein.

The publication of this document as a SIM Guide has been supported by funds of OAS´s project "Implementation of Metrology Infrastructure of the Americas to Support Free Trade and Quality of Life”.

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PURPOSE

This publication has been discussed within the SIM Metrology Working Group of Mass and Related Quantities (SIM MWG7) with the goal of improving the harmonisation of methods for the calibration of Non-Automatic Weighing Instruments (NAWI) within SIM countries. This document provides guidance to national accreditation bodies to set up minimum requirements for the calibration of Non-Automatic Weighing Instruments and gives advice to calibration laboratories to establish practical procedures.

This document contains detailed examples of the estimation of the uncertainty of measurements.

OFFICIAL LANGUAGE

This document was originally written in English, and therefore the English version could be considered as the primary reference, however the Spanish version could be used as a reference as close as possible to the English version.

FURTHER INFORMATION

For further information about this publication, contact the member of the SIM MWG7 of the National Metrology Institute of your country.

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CONTENTS

1 INTRODUCTION 5

2 SCOPE 5

3 TERMINOLOGY AND SYMBOLS 6

4 GENERAL ASPECTS OF THE CALIBRATION 7

4.1 Elements of the Calibration 7

4.2 Test load and indication 8

4.3 Test loads 12

4.4 Indications 15

5 MEASUREMENT METHODS 16

5.1 Repeatability test 16

5.2 Test for errors of indication 17

5.3 Eccentricity test 18

5.4 Auxiliary measurements 18

6 MEASUREMENT RESULTS 19

6.1 Repeatability 19

6.2 Errors of indication 20

6.3 Effect of eccentric loading 21

7 UNCERTAINTY OF MEASUREMENT 21

7.1 Standard uncertainty for discrete values 22

7.2 Standard uncertainty for a characteristic 29

7.3 Expanded uncertainty at calibration 30

7.4 Standard uncertainty of a weighing result 30

7.5 Expanded uncertainty of a weighing result 37

8 CALIBRATION CERTIFICATE 39

8.1 General Information 39

8.2 Information about the calibration procedure 39

8.3 Results of measurement 40

8.4 Additional information 40

9 VALUE OF MASS OR CONVENTIONAL VALUE OF MASS 42

9.1 Value of mass 42

9.2 Conventional value of mass 42

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APPENDICES

(Informative)

A ADVICE FOR ESTIMATION OF AIR DENSITY 44

A1 Formulae for the density of air 44

A2 Variations of parameters constituting the air density 46

A3 Uncertainty of air density 47

B COVERAGE FACTOR K FOR EXPANDED UNCERTAINTY OF MEASUREMENT

49

B1 Objective 49

B2 Basic conditions for the application of k = 2 49

B3 Determining k in other cases 49

C FORMULAE TO DESCRIBE ERRORS IN RELATION TO THE INDICATIONS

51

C1 Objective 51

C2 Functional relations 51

C3 Terms without relation to the readings 56

D SYMBOLS AND TERMS 57

D1 Symbols of general application 57

D2 Locations of important terms and expressions 59

E INFORMATION ON AIR BUOYANCY 62

E1 Density of standard weights 62

E2 Examples for air buoyancy in general 62

E3 Air buoyancy for weights conforming to R111 64

F EFFECTS OF CONVECTION 66

F1 Relation between temperature and time 66

F2 Change of the apparent mass 68

G EXAMPLES 70

G1 Instrument 200 g capacity, scale interval 0,1 mg 70

G2 Instrument 60 kg capacity, multi-interval 74

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1

INTRODUCTION

Nonautomatic weighing instruments are widely used to determine the quantity of a load in terms of mass. While for some applications specified by national legislation, they are subject to legal metrological control – i.e. type approval, verification etc. - there is an increasing need to have their metrological quality confirmed by calibration, e.g. where required by ISO 9001 or ISO/IEC 17025 standards.

2

SCOPE

This document contains guidance for the static calibration of self-indicating, nonautomatic weighing instruments (hereafter called “instrument”), in particular for

1. measurements to be performed, 2. calculation of the measuring results,

3. determination of the uncertainty of measurement, 4. contents of calibration certificates.

The object of the calibration is the indication provided by the instrument in response to an applied load. The results are expressed in units of mass. The value of the load indicated by the instrument will be affected by local gravity, the load’s temperature and density, and the temperature and density of the surrounding air. The uncertainty of measurement depends significantly on properties of the calibrated instrument itself, not only on the equipment of the calibrating laboratory; it can to some extent be reduced by increasing the number of measurements performed for a calibration. This guideline does not specify lower or upper boundaries for the uncertainty of measurement.

It is up to the calibrating laboratory and the client to agree on the anticipated value of the uncertainty of measurement which is appropriate in view of the use of the instrument and in view of the cost of the calibration.

While it is not intended to present one or few uniform procedures the use of which would be obligatory, this document gives general guidance for the establishing of calibration procedures the results of which may be considered as equivalent within the SIM Member Organisations.

Any such procedure must include, for a limited number of test loads, the determination of the error of indication and of the uncertainty of measurement assigned to these errors. The test procedure should as closely as possible resemble the weighing operations that are routinely being performed by the user – e.g. weighing discrete loads, weighing continuously upwards and/or downwards, use of tare balancing function.

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The procedure may further include rules how to derive from the results advice to the user of the instrument with regard to the errors, and assigned uncertainty of measurement, of indications which may occur under normal conditions of use of the instrument, and/or rules on how to convert an indication obtained for a weighed object into the value of mass or conventional value of mass of that object.

The information presented in this guideline is intended to serve, and should be observed by,

1. bodies accrediting laboratories for the calibration of weighing instruments,

2. laboratories accredited for the calibration of nonautomatic weighing instruments,

3. testhouses, laboratories, or manufacturers using calibrated nonautomatic weighing instruments for measurements relevant for the quality of production subject to QM requirements (e.g. ISO 9000 series, ISO 10012, ISO/IEC 17025)

A summary of the main terms and equations used in this document is given in Appendix D2.

3

TERMINOLOGY AND SYMBOLS

The terminology used in this document is mainly based on existing documents:

GUM [1] for terms related to the determination of results and the uncertainty of measurement,

OIML R111 [3] for terms related to the standard weights,

OIML R76 [2] for terms related to the functioning, to the construction, and to the metrological characterisation of nonautomatic weighing instruments.

VIM [7] for terms related to the calibration.

Such terms are not explained in this document, but where they first appear, references will be indicated.

Symbols whose meaning is not self-evident, will be explained where they are first used. Those that are used in more than one section are collected in Appendix D1.

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4

GENERAL ASPECTS OF THE CALIBRATION

4.1 Elements of the calibration

Calibration consists in

1. applying test loads to the instrument under specified conditions, 2. determining the error or variation of the indication, and

3. estimating the uncertainty of measurement to be attributed to the results. 4.1.1 Range of calibration

Unless requested otherwise by the client, a calibration extends over the full weighing range [2] from Zero to the maximum capacity Max. The client may specify a certain part of a weighing range, limited by a minimum load Min′ and the largest load to be weighed Max′, or individual nominal loads, for which he requests calibration.

On a multiple range instrument [2], the client should identify which range(s) shall be calibrated. The preceding paragraph applies to each range separately.

4.1.2 Place of calibration

Calibration is normally performed on the site where the instrument is being used.

If an instrument is moved to another location after the calibration, possible effects from

1. difference in local gravity acceleration, 2. variation in environmental conditions,

3. mechanical and thermal conditions during transportation

are likely to alter the performance of the instrument and may invalidate the calibration. Moving the instrument after calibration should therefore be avoided, unless immunity to these effects of a particular instrument, or type of instrument has been clearly demonstrated. Where this has not been demonstrated, the calibration certificate should not be accepted as evidence of traceability.

4.1.3 Preconditions, preparations

Calibration should not be performed unless

1. the instrument can be readily identified,

2. all functions of the instrument are free from effects of contamination or damage, and functions essential for the calibration operate as intended, 3. presentation of weight values is unambiguous and indications, where

given, are easily readable,

4. the normal conditions of use (air currents, vibrations, stability of the weighing site etc.) are suitable for the instrument to be calibrated,

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5. the instrument is energized prior to calibration for an appropriate period, e.g. as long as the warm-up time specified for the instrument, or as set by the user,

6. the instrument is levelled, if applicable,

7. the instrument has been exercised by loading approximately up to the largest test load at least once, repeated loading is advised.

Instruments that are intended to be regularly adjusted before use should be adjusted before the calibration, unless otherwise agreed with the client. Adjustment should be performed with the means that are normally applied by the client, and following the manufacturer’s instructions where available.

As far as relevant for the results of the calibration, the status of software settings which can be altered by the client should be noted.

Instruments fitted with an automatic zero-setting device or a zero-tracking device [2] should be calibrated with the device operative or not, as set by the client.

For on site calibration the user of the instrument should be asked to ensure that the normal conditions of use prevail during the calibration. In this way disturbing effects such as air currents, vibrations, or inclination of the measuring platform will, so far as is possible, be inherent to the measured values and will therefore be included in the determined uncertainty of measurement.

4.2 Test load and indication

4.2.1 Basic relation between load and indication

In general terms, the indication of an instrument is proportional to the force exerted by an object of mass m on the load receptor:

I ~mg

(

1−

ρ

a

ρ

)

(4.2.2-1)

with g local gravity acceleration

a

ρ

density of the surrounding air ρ density of the object

The terms in the brackets account for the reduction of the force due to the air buoyancy of the object.

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4.2.2 Effect of air buoyancy

It is state of the art to use standard weights that have been calibrated to the conventional value of massmc1, for the adjustment and/or the calibration of weighing instruments. The adjustment is performed such that effects of g and of the actual buoyancy of the standard weight mcs are included in the adjustment factor. Therefore, at the moment of the adjustment the indication Is is

cs s m

I = (4.2.2-1)

This adjustment is, of course, performed under the conditions characterized by the actual values ofgs, ρs ≠ ρc, and ρas ≠ ρ0, identified by the suffix “s”,and is valid only under these conditions. For another body with ρ ≠ρs, weighed on the same instrument but under different conditions: g ≠gs y ρa ≠ ρas the indication is in general (neglecting terms of 2nd or higher order):

(

s

) (

{

a

)

(

s

)

(

a as

)

s

}

c g g

m

I = / 1− ρ −ρ0 1ρ−1 ρ − ρ −ρ /ρ (4.2.2-3)

If the instrument is not displaced, there will be no variation ofg, so g gs =1. This is assumed hereafter.

The formula simplifies further in situations where some of the density values are equal:·

a) weighing a body in the reference air density: ρa0, then

(

)

{

a as s

}

c

m

I = 1− ρ −ρ /ρ (4.2.2-4)

b) weighing a body of the same density as the adjustment weight:

s

ρ

ρ= , then again (as in case a)

(

)

{

a as s

}

c

m

I = 1− ρ −ρ /ρ (4.2.2-5)

c) weighing in the same air density as at the time of adjustment:

as a ρ ρ = , then

(

)

(

)

{

a s

}

c m I = 1− ρ −ρ0 1 ρ−1ρ (4.2.2-6) 1

The conventional value of mass mcof a body has been defined in [3] as the numerical value of mass m of a weight of reference density ρc= 8000 kg/m³ which balances that body at 20 °C in air of densityρ0:

( ) ( )

{ −ρ ρ −ρ ρ }

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Figure 4.2-1 shows examples for the magnitude of the relative changes

(

s

)

s

s I I I

I

I/ = − /

∆ for an instrument adjusted with standard weights of ρsc, when calibrated with standard weights of different but typical density.

Line ▲ is valid for a body of ρ = 7 810 kg/m³, weighed in ρaas Line

×

is valid for a body of ρ = 8 400 kg/m³, weighed in ρaas Line

is valid for a body of ρ =ρsc after adjustment inρas0

It is obvious that under these conditions, a variation in air density has a far greater effect than a variation in the body’s density.

Further information is given on air density in Appendix A, and on air buoyancy related to standard weights in Appendix E.

4.2.3 Effects of convection

Where weights have been transported to the calibration site they may not have the same temperature as the instrument and its environment. Two phenomena should be noted in this case:

An initial temperature difference∆T0 may be reduced to a smaller value∆T

by acclimatisation over a time ∆t; this occurs faster for smaller weights than for larger ones.

When a weight is put on the load receptor, the actual difference ∆T will produce an air flow about the weight leading to parasitic forces which result in an apparent change ∆mconv on its mass. The sign of ∆mconv is normally opposite to the sign of ∆T, its value is greater for large weights than for small ones.

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The relations between any of the quantities mentioned: ∆T0, ∆t, ∆T, m and

conv

m

∆ are nonlinear, and they depend on the conditions of heat exchange between the weights and their environment – see [5].

Figure 4.2-2 gives an impression of the magnitude of the apparent change in mass in relation to a temperature difference, for some selected weight values.

This effect should be taken into account by either letting the weights accomodate to the extent that the remaining change ∆mconv is negligible in view of the uncertainty of the calibration required by the client, or by considering the possible change of indication in the uncertainty budget. The effect may be significant for weights of high accuracy, e.g. for weights of class E2 or F1 in R 111 [3].

More detailed information is given in Appendix F.

4.2.4 Reference value of mass

The general relations (4.2.2-3) to (4.2.2-6) apply also if the “body weighed” is a standard weight used for calibration.

To determine the errors of indication of an instrument, standard weights of known conventional value of mass mcCal are applied. Their density ρCal is normally different from the reference value ρc and the air density ρaCal at the time of calibration is normally different from ρ0.

The error E of indication is

ref

m I

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where mref is a conventional true value of mass, further called reference value of mass. Due to effects of air buoyancy, convection, drift and others which may lead to minor correction terms δmx, mref is not exactly equal to mcCal:

... m m m m m

mref = cCalBconvD +δ (4.2.4-2) The correction for air buoyancy

δ

mB is affected by values of ρs and ρas, that were valid for the adjustment but are not normally known. It is assumed that weights of the reference densityρsc have been used. (4.2.2-3) then gives the general expression for the correction

(

)(

) (

)

[

aCal Cal c aCal as c

]

cCal

B m

m ρ ρ ρ ρ ρ ρ ρ

δ =− − 0 1 −1 + − (4.2.4-3)

For the air density ρas two situations are considered:

A The instrument has been adjusted immediately before the calibration, so

aCal as ρ

ρ = . This simplifies (4.2.4-3) to:

(

aCal

)(

Cal c

)

cCal

B m

m ρ ρ ρ ρ

δ =− − 0 1 −1 (4.2.4-4)

B The instrument has been adjusted independent of the calibration, in unknown air density ρas for which a reasonable assumption should be made.

B1 For on-site calibrations,ρas may be expected to be similar to ρaCal, with the possible difference δρasaCal −ρas. (4.2.4-3) is then modified to

(

)(

)

[

aCal Cal c as c

]

cCal B m m ρ ρ ρ ρ δρ ρ δ =− − 0 1 −1 + (4.2.4-5)

B2 A simple, straightforward assumption could be ρas0, then

(

aCal

)

Cal

cCal

B m

m ρ ρ ρ

δ =− − 0 / (4.2.4-6)

See also Appendices A and E for further information. The other correction terms are dealt with in section 7.

The suffix “Cal” will from now on be omitted unless where necessary to avoid confusion.

4.3 Test loads

Test loads should preferably consist of standard weights that are traceable to the SI unit of mass. Other test loads may be used, however, for tests of a comparative nature – e.g. test with eccentric loading, repeatability test – or for the mere loading of an instrument – e.g. preloading, tare load that is to be balanced, substitution load.

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4.3.1 Standard weights

The traceability of weights to be used as standards shall be accomplished by calibration2 consisting of

1. determination of the actual conventional value of mass mc and/or the correction δmc to its nominal value m: δmc =mc −m, together with the expanded uncertainty of the calibration U95, or

2. confirmation that mc is within specified maximum permissible errors

mpe: m

(

mpe−U95

)

<mc< m +

(

mpe−U95

)

The standards should further satisfy the following requirements to the extent as appropriate in view of their accuracy:

3. density ρs sufficiently close to ρC = 8 000 kg/m³

4. surface finish suitable to prevent a change in mass through contamination by dirt or adhesion layers

5. magnetic properties such that interaction with the instrument to be calibrated is minimized.

Weights that comply with the relevant specifications of the International Recommendation OIML R 111 [3] should satisfy all these requirements.

The maximum permissible errors, or the uncertainties of calibration of the standard weights shall be compatible with the scale interval d [2] of the instrument and/or the needs of the client with regard to the uncertainty of the calibration of his instrument. 4.3.2 Other test loads

For certain applications mentioned in 4.3, 2nd sentence, it is not essential that the conventional value of mass of a test load is known. In these cases, loads other than standard weights may be used, with due consideration of the following:

1. shape, material, composition should allow easy handling,

2. shape, material, composition should allow the position of the centre of gravity to be readily estimated,

3. their mass must remain constant over the full period they are in use for the calibration,

4. their density should be easy to estimate,

5. loads of low density (e.g. containers filled with sand or gravel), may require special attention in view of air buoyancy.

Temperature and barometric pressure may need to be monitored over the full period the loads are in use for the calibration.

2

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4.3.3 Use of substitution loads

A test load the conventional value of mass of which is essential, should be made up entirely of standard weights. But where this is not possible, any other load which satisfies 4.3.2 may be used for substitution. The instrument under calibration is used as comparator to adjust the substitution load Lsub so that it brings about approximately the same indication I as the corresponding load LSt made up of standard weights.

A first test load LT1 made up of standard weights mc1 is indicated as:

( ) ( )

LSt I mc1

I = (4.3.3-1)

After removing LSt a substitution load Lsub1 is put on and adjusted to give approximately the same indication:

(

Lsub1

) ( )

I mc1 I ≈ (4.3.3-2) so that

(

1

) ( )

1 1 1 1 1 m I L I m m I Lsub = c + subc = c +∆ (4.3.3-3)

The next test load LT2 is made up by adding mc1

1 1 1 1 2 L m 2m I LT = sub + c = c +∆ (4.3.3-4) 1 c

m is again replaced by a substitution load of ≈ Lsub1 with adjustment to ≈ I

( )

LT2 . The procedure may be repeated, to generate test loads LT3, ...LTn :

1 2 1 1+∆ +∆ + +∆ − = c n Tn nm I I I L K (4.3.3-5)

The value of LTnis taken as the conventional value of mass mc of the test load. With each substitution step however, the uncertainty of the total test load increases substantially more than if it were made up of standard weights only, due to the effects of repeatability and resolution of the instrument. – cf. also 7.1.2.63.

3

Example: for an instrument with Max = 5000 kg, d = 1 kg, the standard uncertainty of 5 t standard weights may be 200 g, while the standard uncertainty of a test load made up of 1 t standard weights and 4 t substitution load, will be about 2

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4.4 Indications

4.4.1 General

Any indication I related to a test load is basically the difference of the indications

L

I under load andI0 at no-load:

0

I I

I = L − (4.4.1-1)

It is to be preferred to record the no-load indications together with the load indications for any test measurement. However, recording the no-load indications may be redundant where a test procedure calls for the setting to zero of any no-load indication which is not = zero of itself, before a test no-load is applied.

For any test load, including no load, the indication I of the instrument is read and recorded only when it can be considered as being stable. Where high resolution of the instrument, or environmental conditions at the calibration site prevent stable indications, an average value should be recorded together with information about the observed variability (e.g. spread of values, unidirectional drift).

During calibration tests, the original indications should be recorded, not errors or variations of the indication.

4.4.2 Resolution

Indications are normally obtained as integer multiples of the scale interval d.

At the discretion of the calibration laboratory and with the consent of the client, means to obtain indications in higher resolution than in d may be applied, e.g. where compliance to a specification is checked and smallest uncertainty is desired. Such means may be:

1. switching the indicating device to a smaller scale interval dT < d (“service mode”).

In this case, the indication Ix is then obtained as integer multiple of dT. 2. applying small extra test weights in steps of dT =d 5 or d 10 to determine more precisely the load at which an indication changes unambiguously from I′ to I′+d. (“changeover point method”). In this case, the indication I’ is recorded together with the amount ∆L of the n additional small test weights necessary to increase I′ by one d.

The indication IL is

T

L I d L I d nd

I = ' + 2−∆ = ′+ 2− (4.4.2-1)

Where the changeover point method is applied, it is advised to apply it for the indications at zero as well where these are recorded.

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5

MEASUREMENT METHODS

Tests are normally performed to determine • the repeatability of indications, • the errors of indications,

• the effect of eccentric application of a load on the indication.

A Calibration Laboratory deciding on the number of measurements for its routine calibration procedure, should consider that in general, a larger number of measurements tends to reduce the uncertainty of measurement but to increase the cost.

Details of the tests performed for an individual calibration may be fixed by agreement of the client and the Calibration Laboratory, in view of the normal use of the instrument. The parties may also agree on further tests or checks which may assist in evaluating the performance of the instrument under special conditions of use. Any such agreement should be consistent with the minimum numbers of tests as specified in the following sections.

5.1 Repeatability test

The test consists in the repeated deposition of the same load on the load receptor, under identical conditions of handling the load and the instrument, and under constant test conditions, both as far as possible.

The test load(s) need not be calibrated nor verified, unless the results serve for the determination of errors of indication as per 5.2. The test load should, as far as possible, consist of one single body.

The test is performed with at least one test load LT which should be selected in a reasonable relation to Max and the resolution of the instrument, to allow an appraisal of the intrument's performance. For instruments with a constant scale interval d a load of0,5Max≤LT ≤Max is quite common; this is often reduced for instruments where LT >0,5Max would amount to several 1000 kg. For multi-interval instruments [2] a load close to Max1may be preferred. A special value of

T

L may be agreed between the parties where this is justified in view of a specific application of the instrument.

The test may be performed at more than one test point, with test loads LTj,

L

k

j≤

1 with kL = number of test points.

Prior to the test, the indication is set to zero. The load is to be applied at least 5 times, and at least 3 times where LT ≥ 100 kg.

Indications ILi are recorded for each deposition of the load. After each removal of the load, the indication should at least be checked for showing zero, and may be reset to zero if it does not; recording of the no-load indications I0i is advisable as per 4.4.1. In addition, the status of the zero device if fitted is recorded.

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5.2 Test for errors of indication

1. This test is performed with kL ≥ 5 different test loads LTj, 1 ≤ j ≤ kL, distributed fairly evenly over the normal weighing range4or at individual test points agreed upon as per 4.1.2.

The purpose of this test is an appraisal of the performance of the instrument over the whole weighing range.

Where a significantly smaller range of calibration has been agreed to, the number of test loads may be reduced accordingly, provided there are at least 3 test points including Min′ and Max′, and the difference between two consecutive test loads is not greater than 0,15Max.

It is necessary that test loads consist of appropriate standard weights, or of substitution loads as per 4.3.3.

Prior to the test, the indication is set to zero. The test loads LTj are normally applied once in one of these manners:

1. increasing by steps with unloading between the separate steps – corresponding to the majority of uses of the instruments for weighing single loads,

2. continuously increasing by steps – similar to 1; may include creep effects in the results, reduces the amount of loads to be moved on and off the load receptor as compared to 1,

3. continuously increasing and decreasing by steps – procedure prescribed for verification tests in [2], same comments as for 2,

4. continuously decreasing by steps starting from Max- simulates the use of an instrument as hopper weigher for subtractive weighing, same comments as for 2.

On multi-interval instruments – see [2], the methods above may be modified for load steps smaller than Max, by applying increasing and/or decreasing tare loads, operating the tare balancing function, and applying a test load of close to but not more than Max1 to obtain indications with d1.

Further tests may be performed to evaluate the performance of the instrument under special conditions of use, e.g. the indication after a tare balancing operation, the variation of the indication under a constant load over a certain time, etc.

The test, or individual loadings, may be repeated to combine the test with the repeatability test under 5.1.

Indications ILj are recorded for each load. After each removal of a load, the indication should at least be checked for showing zero, and may be reset to zero if

4

Examples for target values::

L

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it does not; recording of the no-load indications I0j as per 4.4.1.

5.3 Eccentricity test

The test consists in placing a test load Lecc in different positions on the load receptor in such a manner that the centre of gravity of the load takes the positions as indicated in Figure 5.3-1 or equivalent positions, as closely as possible.

Fig. 5.3-1 Positions of load for test of eccentricity 1. Centre 2. Front left 3. Back left 4. Back right 5. Front right

The test load Lecc should be at leastMax 3, or at least Min′+

(

Max′−Min′

)

3 for a reduced weighing range. Advice of the manufacturer, if available, and limitations that are obvious from the design of the instrument should be considered – e.g. see OIML R76 [2] for weighbridges.

The test load need not be calibrated nor verified, unless the results serve for the determination of errors of indication as per 5.2.

Prior to the test, the indication is set to Zero. The test load is first put on position 1, is then moved to the other 4 positions in arbitrary order, and may at last be again put on position 1.

Indications ILi are recorded for each position of the load. After each removal of the load, the zero indication should be checked and may, if appropriate, be reset to zero; recording of the no-load indications I0j as per 4.4.1.

5.4 Auxiliary measurements

The following additional measurements or recordings are recommended, in particular where a calibration is intended to be performed with the lowest possible uncertainty.

In view of buoyancy effects – cf. 4.2.2:

The air temperature in reasonable vicinity to the instrument should be measured, at least once during the calibration. Where an instrument is used in a controlled environment, the span of the temperature variation should be noted, e.g. from a temperature graph, from the settings of the control device etc.

Barometric pressure or, by default, the altitude above sea-level of the site may be useful.

In view of convection effects – cf 4.2.3:

Special care should be taken to prevent excessive convection effects, by observing a limiting value for the temperature difference between standard weights and instrument, and/or recording an acclimatisation time that has been accomplished. A thermometer kept inside the box with standard weights may be helpful, to check

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the temperature difference.

In view of effects of magnetic interaction:

On high resolution instruments a check is recommended to see if there is an observable effect of magnetic interaction. A standard weight is weighed together with a spacer made of non-metallic material (e.g. wood, plastic), the spacer being placed on top or underneath the weight to obtain two different indications.

If the difference of these two indications is different from zero, this should be mentioned as a warning in the calibration certificate.

6

MEASUREMENT RESULTS

The formulae in chapters 6 and 7 are intended to serve as elements of a standard scheme for an equivalent evaluation of the results of the calibration tests. Where they are being applied unchanged as far as applicable, no further description of the evaluation is necessary.

It is not intended that all of the formulae, symbols and/or indices are used for presentation of the results in a Calibration Certificate.

The definition of an indication I as given in 4.4 is used in this section.

6.1 Repeatability

From the n indications Iji for a given test load LTj, the standard deviation sj is calculated

( )

(

)

= − − = n i j ji j I I n I s 1 2 1 1 (6.1-1) with

= = n i ji j I n I 1 1 (6.1-2)

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6.2 Errors of indication

6.2.1 Discrete values

For each test load LTi, the error of indication is calculated as follows:

refj j j I m

E = − (6.2-1a)

Where an indication Ij is the mean of more than one reading, Ij is understood as being the mean value as per (6.1-2).

ref

m is the reference weight or “true value” of the load. – cf. 4.3.1, 4.3.3. The reference weight is

either the nominal value m of the load,

j refj m

m = (6.2-2)

or its actual value mc

(

j cj

)

cj

refj m m m

m = = +δ (6.2-3)

Where a test load was made up of more than 1 weight, mj is replaced by

(

)

j m

and δmcj is replaced by

(

)

j c m

δ in the formulae above.

Where an error and/or indication is listed or used further in relation to the test load, it should always be presented in relation to the nominal value m of the load, even if the actual value of mass of the test load has been used. In such a case, the error remains unchanged, while the indication is modified by

( )

m I

( )

mc mc

I = ′ −δ (6.2-4)

withI′ being the (interim) indication determined when mc was applied. (6.2-1a) then takes the form

(

j cj

)

j j

j

j I m I m m

E = − = ′ −δ − (6.2-1b)

6.2.2 Characteristic of the weighing range

In addition, or as an alternative to the discrete values Ij, Ej, a characteristic, or calibration curve may be determined for the weighing range, which allows to estimate the error of indication for any indication I within the weighing range.

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A function

( )

I f

Eappr = (6.2-5)

may be generated by an appropriate approximation which should in general, be based on the “least squares” approach:

( )

(

)

2 =

− 2 = j j j f I E v minimum (6.2-6) with j v = residual f = approximation function The approximation should further

• take account of the uncertainties u

( )

Ej of the errors,

• use a model function that reflects the physical properties of the instrument, e.g. the form of the relation between load and its indication

( )

L g

I = ,

• include a check whether the parameters found for the model function are mathematically consistent with the actual data.

It is assumed that for any mj the error Ej remains the same if the actual indication Ij is replaced by its nominal value Ij. The calculations to evaluate (6.2-6) can therefore be performed with the data sets mj, Ej, or Ij, Ej.

Appendix C offers advice for the selection of a suitable approximation formula and for the necessary calculations.

6.3 Effect of eccentric loading

From the indications Ii obtained in the different positions of the load as per 5.3, the differences ∆Iecc are calculated

1

I I Iecci = i

∆ (6.3-1)

Where the test load consisted of standard weight(s), the errors of indication may be calculated instead: i ecci I m E = − (6.3-2)

7

UNCERTAINTY OF MEASUREMENT

In this section and the ones that follow, there are terms of uncertainty assigned to small corrections, which are proportional to a specified mass value or to a specified indication. For the quotient of such an uncertainty divided by the related value of mass or indication, the abbreviated notation wˆ will be used.

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Example: let

(

m

)

mu

(

corr

)

u δ corr = (7-1)

with the dimensionless term u

(

corr

)

, then

(

m

) (

u corr

)

corr = (7-2)

Accordingly, the related variance will be denoted by wˆ2

(

mcorr

)

and the related expanded uncertainty by Wˆ

(

mcorr

)

.

7.1 Standard uncertainty for discrete values

The basic formula for the calibration is

ref

m I

E= − (7.1-1)

with the variances

( )

E u

( )

I u

( )

mref

u2 = 2 + 2 (7.1-2)

Where substitution loads are employed – see 4.3.3 -mref is replaced by LTn in both expressions.

The terms are expanded further hereafter. 7.1.1 Standard uncertainty of the indication

To account for sources of variability of the indication, (4.4.1-1) is amended by correction terms δIxx as follows:

0 0 dig ecc rep digL L I I I I I I I = +δ +δ +δ − −δ (7.1.1-1)

All these corrections have the expectation value zero. Their standard uncertainties are:

7.1.1.1 δIdig0 accounts for the rounding error of no-load indication. Limits are ±d0 2 or 2

T

d

± as applicable; rectangular distribution is assumed, therefore

(

I 0

)

d0

( )

2 3 uδ dig = (7.1.1-2a) or

(

Idig0

)

dT

( )

2 3 uδ = (7.1.1-2b) respectively.

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Note 1: cf. 4.4.2 for significance of dT.

Note 2: on an instrument which has been type approved to OIML R76 [2], the rounding error of a zero indication after a zero-setting or tare balancing operation is limited to ±d0 4, therefore

(

I 0

)

d0

( )

4 3

dig = (7.1.1-2c)

7.1.1.2 δIdigL accounts for the rounding error of indication at load. Limits are ±dI 2 or 2

T

d

± as applicable; rectangular distribution to be assumed, therefore

(

IdigL

)

dI 2 3

uδ = (7.1.1-3a

or

(

IdigL

)

dT 2 3

uδ = (7.1.1-3b)

Note: on a multi-interval instrument,dI varies with I !

7.1.1.3 δIrep accounts for the error due to imperfect repeatability; normal distribution is assumed, estimated

( ) ( )

Irep s Ij

uδ = (7.1.1-5)

with s

( )

Ij as per 6.1.

Where an indication Ij is the mean of n readings, the corresponding standards uncertainty is

( ) ( )

I s I n

rep = j (7.1.1-6)

Where only one repeatability test has been performed, this standard deviation may be considered as being representative for all indications of the instrument in the weighing range considered.

Where several sj (sj =s

( )

Ij in abbreviated notation) have been determined with different test loads, the greater value of sj for the two test points enclosing the indication whose error has been determined, should be used.

Where it can be established that the values of sj determined at different test loads

Tj

L , are in functional relation to the load, this function may be applied to combine the sj values into a “pooled” standard deviation spool.

Examples for such functions are = j s const (7.1.1-7)

(

)

2 2 2 0 2 Max L s s sj = + r Tj (7.1.1-8) The components 2 0 s and 2 r

s have to be determined either by a graph or by calculation.

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Note: For a standard deviation reported in a calibration certificate, it should be clear whether it is related to a single indication or to the mean of n indications. 7.1.1.4 δIecc accounts for the error due to off-centre position of the centre of gravity of a

test load. This effect may occur where a test load is made up of more than one body. Where this effect cannot be neglected, an estimate of its magnitude may be based on these assumptions:

the differences determined by (6.3-1) are proportional to the distance of the load from the centre of the load receptor and to the value of the load;

the eccentricity of the effective centre of gravity of the test load is no more than12of the value at the eccentricity test.

While there may be instruments on which the effect of eccentric loading is even greater at other angles than those where the test loads have been applied, based on the largest of the differences determined as per 6.3, the effect is estimated to be

(

)

{

I L

}

I

Iecc ecci 2 ecc

max ,

∆ ≤

δ (7.1.1-9)

Rectangular distribution is assumed, so the standard uncertainty is

(

)

(

2 3

)

max ,i ecc ecc ecc I I L I u δ = ∆ (7.1.1-10)

or, in relative notation,

(

)

(

2 3

)

ˆ max ,i ecc ecc ecc I L I wδ = ∆ (7.1.1-11)

7.1.1.5 The standard uncertainty of the indication is normally obtained by

( )

2 2 2

( )

2

(

)

2 0 2 ˆ 12 12 d s I w I I d I u = + I + + δ ecc (7.1.1-12)

Note 1: the uncertainty u

( )

I is = constant only where s = constant and no eccentricity error has to be considered.

Note 2: the first two terms on the right hand side may have to be modified in special cases as mentioned in 7.1.1.1 and 7.1.1.2.

7.1.2 Standard uncertainty of the reference mass From 4.2.4 and 4.3.1 the reference value of mass is:

K m m m m m m mref = cBDconv +δ (7.1.2-1)

The rightmost term stands for further corrections that may in special conditions be necessary to apply, it is not further considered hereafter.

The corrections and their standard uncertainties are:

7.1.2.1 δmc is the correction to m to obtain the actual conventional value of mass mc; given in the calibration certificate for the standard weights, together with the uncertainty of calibration U and the coverage factor k. The Standard uncertainty is

(

m

)

U k

u δ c = (7.1.2-2)

Where the Standard weight has been calibrated to specified tolerances Tol, e.g. to the mpe given in R 111, and where it is used its nominal value m, then δmc= 0,

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and rectangular distribution is assumed, therefore

(

m

)

Tol 3

u

δ

c = (7.1.2-3)

Where a test load consists of more than one standard weight, the standard uncertainties are summed up arithmetically not by sum of squares, to account for assumed correlations

For test loads partially made up of substitution loads see 7.1.2.6 Note 1: cf. 6.2.1 for use of mc or m.

Note 2: Where conformity of the standard weight(s) to R 111 is established, (7.1.2-3) may be modified by replacing Tol by mpe. For weights of m ≥ 0,1 kg the quotient mpe/m is constant for all weights belonging to the same accuracy class,

mpe = cclassm with cclass from Table 7.1-1.

(7.1.2-3) may then be used in the form

(

mc

)

mcclass 3

u

δ

= (7.1.2-3a)

or as relative standard uncertainty

(

)

3

ˆ mc cclass

w

δ

= (7.1.2-3b)

Table 7.1-1 Quotient cclass =mpe m for standard weights m ≥ 100 g according to R 111 [3]

Class class c × 106 E1 0,5 E2 1,6 F1 5 F2 16 M1 50 M2 160 M3 500

For weights of nominal value of 2 x 10n of the following classes: E2, F2 and M2, the value of cclass× 106should be substituted by 1,5, 15 and 150 respectively.

7.1.2.2

δ

mB is the correction for air buoyancy as introduced in 4.2.4. The value depends on the density ρ of the calibration weight, on the assumed range of air density ρa, and on the adjustment of the instrument – cf cases A and B in 4.2.4.

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Case A:

(

a

)(

c

)

B m m ρ ρ ρ ρ δ =− − 0 1 −1 (7.1.2-4)

with the relative standard uncertainty from

( )

( )(

) (

) ( )

2 2 4 0 2 2 2 1 1 ˆ m u

ρ

ρ

ρ

ρ

ρ

u

ρ

ρ

w B = ac + a − (7.1.2-5) Case B1:

(

)(

)

[

a c as c

]

cCal B m m ρ ρ ρ ρ δρ ρ δ =− − 0 1 −1 + (7.1.2-6)

with the relative standard uncertainty from

( )

( )(

) (

) ( )

2 2 4 2

(

)

2 0 2 2 2 1 1 ˆ mB u a c a u u as c w =

ρ

ρ

ρ

+

ρ

ρ

ρ

ρ

+

δρ

ρ

(7.1.2-7) Case B2:

(

ρ ρ

)

ρ δmB =−m a0 (7.1.2-8)

with the relative standard uncertainty from

( )

( )

(

) ( )

2 2 4 0 2 2 2 ˆ m u

ρ

ρ

ρ

ρ

u

ρ

ρ

w B = a + a − (7.1.2-9)

As far as values for ρ, u

( )

ρ

, ρa and u

( )

ρa , are known, these values should be used to determine wˆ

( )

mB .

The density ρ and its standard uncertainty may in the absence of such information, be estimated according to the state of the art. Appendix E1 offers internationally recognized values for common materials used for standard weights.

The air density ρa and its standard uncertainty can be calculated from temperature and barometric pressure if available (the relative humidity being of minor influence), or may be estimated from the altitude above sea-level.

For the difference δρas (Case B1), zero may be assumed with an appropriate uncertainty u

(

δρas

)

for which a limit ∆ρas should be estimated taking into account the variability of barometric pressure and temperature at the site, over a longer period of time.

A simple approach may be to use the same estimates forρa and ρas the same uncertainty for both values.

Appendix A offers several formulae, and information about expected variances. Appendix E offers values of wˆ

( )

mB for some selected combinations of values for ρ and ρa. For case A calibrations, the values are mostly negligible.

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For case B calibrations, it may mostly be advisable to not apply a correction

δ

mB

but to calculate the uncertainty based onρ and on ρa0 ±∆ρa

Where conformity of the standard weights to R 111[3], is established, and no information on ρ and ρa, is at hand, recourse may be taken to section 10 of R 1115. No correction is applied, and the relative uncertainties are

for case A,

( )

(

4 3

)

ˆ mB mpe m

w ≈ ( 7.1.2-5a)

For cases B1 and B2,

( )

(

0,1

(

4

)

)

3

ˆ mB 0 c mpe m

w ≈

ρ

ρ

+ (7.1.2-9a)

From the requirement in footnote 5, these limits can be derived for ρ: For class E2:

ρ

ρ

c ≤ 200 kg/m³, and for class F1:

ρ

ρ

c ≤ 600 kg/m³.

Note: Due to the fact that the density of materials used for standard weights is normally closer to ρc than the R111 limits would allow, the last 2 formulae may be considered as upper limits for wˆ

( )

mB . Where a simple comparison of these values with the resolution of the instrument

(

1nM =d Max

)

shows they are small enough, a more elaborate calculation of this uncertainty component based on actual data, may be superfluous.

7.1.2.3

δ

mD is a correction for a possible drift of mc since the last calibration. A limiting value D is best assumed, based on the difference in mc evident from consecutive calibration certificates of the standard weights.

In the absence of such information, D may be estimated in view of the quality of the weights, and frequency and care of their use, to a multiple of their expanded uncertainty U

( )

mc :

( )

c DU m

k

D= (7.1.2-10)

where kD may be chosen from 1 to 3.

It is not advised to apply a correction but to assume even distribution within ±D

(rectangular distribution). The standard uncertainty is then

(

m

)

D 3

D = (7.1.2-11)

Where a set of weights has been calibrated with a standardized expanded relative uncertainty Wˆ

( )

mc , it may be convenient to introduce a relative limit value for drift

rel D m

D = and a relative uncertainty for drift

( )

3 ˆ

( )

3

ˆ mD Drel kDW m

w = = (7.1.2-12)

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For weights conforming to R111 [3], the estimate may be D≤mpe, or Drel ≤cclass – see Table 7.1-1

7.1.2.4 δmconv is a correction for convection effects as per 4.2.3. A limiting value ∆mconv

may be taken from Appendix F, depending on a known difference in temperature

T

∆ and on the mass of the standard weight.

It is not advised to apply a correction but to assume even distribution within

conv

m

± . The standard uncertainty is then

(

mconv

)

mconv 3

u

δ

=∆ (7.1.2-13)

7.1.2.5 The standard uncertainty of the reference mass is obtained from – cf. 7.1.2

( )

mref u

(

mc

)

u

(

mB

)

u

(

mD

)

u

(

mconv

)

u2 = 2

δ

+ 2

δ

+ 2

δ

+ 2

δ

(7.1.2-14)

with the contributions from 7.1.2.1 to 7.1.2.4

As an example the terms are specified for a case A calibration with standard weights of m ≥ 0,1 kg conforming to R111, used with their nominal values:

( )

3 48 3

(

)

3

ˆ2 mref cclass2 cclass2 cclass2 mconv m 2

w = + + + ∆ (7.1.2-14a)

7.1.2.6 Where a test load is partially made up of substitution loads as per 4.3.3, the standard uncertainty for the sum LTn =nmc1 +∆I1 +∆I2 +K+∆In1 is given by the

following expression:

( )

( )

[

( )

( )

( )

1

]

2 2 2 1 2 1 2 2 2 2 − + + + + = c n Tn n u m u I u I u I L u K (7.1.2-15)

with u

( )

mc1 =u

( )

mref from 7.1.2.5, and u

( )

Ij from 7.1.1.5 for I = I

( )

LTj

Note: the uncertainties u

( )

Ij have to be included also for indications where the substitution load has been so adjusted that the corresponding ∆I becomes zero! Depending on the kind of the substitution load, it may be necessary to add further uncertainty contributions:

for eccentric loading as per 7.1.1.4 to some or all of the actual indications I

( )

LTj ;

for air buoyancy of the substitution loads, where these are made up of low density materials (e.g. sand, gravel) and the air density varies significantly over the time the substitution loads are in use.

Where u

( )

Ij = const, the expression simplifies to

( )

L n u

( )

m

[

(

n

) ( )

u I

]

u2 Tn = 2 2 c1 +2 −1 2 (7.1.2-16)

7.1.3 Standard uncertainty of the error

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appropriate, calculated from

( )

( )

(

)

(

c

)

(

B

)

(

D

)

(

conv

)

ecc I m u m u m u m u I u I s d d E u

δ

δ

δ

δ

δ

2 2 2 2 2 2 2 2 0 2 12 12 + + + + + + + = (7.1.3-1a) or, where relative uncertainties apply, from

( )

( )

( )

( )

( )

( )

{

c B D

}

ref

(

conv

)

ecc I m u m m w m w m w I I w I s d d E u δ 2 2 2 2 2 2 2 2 2 2 0 2 ˆ ˆ ˆ ˆ 12 12 + + + + + + + = (7.1.3-1b)

All input quantities are considered to be uncorrelated, therefore covariances are not considered.

The index “ j”has been omitted. Where the last terms in (7.1.3-1a, b) are small compared to the first 3 terms, the uncertainty of all errors determined over the weighing range is likely to be quite similar. If this is not the case, the uncertainty has to be calculated individually for each indication.

In view of the general experience that errors are normally very small compared to the indication, or may even be zero, in (7.1.3-1a, b) the values for mref and I may be replaced by I.

The terms in (7.1.3-1a, b) may then be grouped into a simple formula which better reflects the fact that some of the terms are absolute in nature while others are proportional to the indication:

( )

2 2 2 2 I E u =α +β (7.1.3-2)

Where (7.1.1-7) or (7.1.1-8) applies to the standard deviation determined for the calibrated instrument, the corresponding terms are of course included in (7.1.3-2).

7.2 Standard uncertainty for a characteristic

Where an approximation is performed to obtain a formula E= f

( )

I for the whole weighing range as per 6.2.2, the standard uncertainty of the error per 7.1.3 has to be modified to be consistent with the method of approximation. Depending on the model function, this may be

a single variance u2appr which is added to (7.1.3-1), or

a set of variances and covariances which include the variances in (7.1.3-1). The calculations should also include a check whether the model function is mathematically consistent with the data sets Ej, Ij, u

( )

Ej .

The 2

minχ , approach which is similar to the least squares approach, is proposed for approximations. Details are given in Appendix C

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7.3 Expanded uncertainty at calibration

The expanded uncertainty of the error is

( )

E ku

( )

E

U = (7.3-1)

The coverage factor k, should be chosen such that the expanded uncertainty corresponds to a coverage probability of approximately 95 %.

The value k =2, corresponding to a 95,5% probability, applies where

a) a normal (Gaussian) distribution can be attributed to the error of indication, and

b) the standard uncertainty u

( )

E is of sufficient reliability (i.e. it has a sufficient number of degrees of freedom).

Appendix B2 offers additional information to these conditions, and Appendix B3 advises how to determine the factor k where one or both of them are not met.

It is acceptable to determine only one value of k, for the “worst case” situation identified by experience, which may be applied to the standard uncertainties of all errors of the same weighing range.

7.4 Standard uncertainty of a weighing result

The user of an instrument should be aware of the fact that in normal usage of an instrument that has been calibrated, the situation is different from that at calibration in some if not all of these aspects:

1. the indications obtained for weighed bodies are not the ones at calibration,

2. the weighing process may be different from the procedure at calibration:

a. certainly only one reading for each load, not several readings to obtain a mean value,

b. reading to the scale interval d, of the instrument, not with higher resolution,

c. loading up and down, not only upwards – or vice versa,

d. load kept on load receptor for a longer time, not unloading after each loading step – or vice versa,

e. eccentric application of the load, f. use of tare balancing device,

etc.

3. the environment (temperature, barometric pressure etc.) may be different,

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4. on instruments which are not readjusted regularly e.g. by use of a built-in device, the adjustment may have changed, due to ageing or to wear and tear.

Unlike the items 1 to 3, this effect is usually depending on the time that has elapsed since the calibration, it should therefore be considered in relation to a certain period of time, e.g. for one year or the normal interval between calibrations.

In order to clearly distinguish from the indications I obtained during calibration, the weighing results obtained when weighing a load L on the calibrated instrument, these terms and symbols are introduced:

R = reading, any indication obtained after the calibration;

W = weighing result, reading corrected for the errorE.

R is understood as a single reading in normal resolution (multiple of d), with corrections to be applied as applicable.

For a reading taken under the same conditions as those prevailing at calibration, for a load well centered on the load receptor, only corrections to account for points 2a and 2b above apply. The result may be denominated as the weighing result under the conditions of the calibration W*:

(

R R

)

E

R R

R

W*= +δ digLrep0dig0 − (7.4-1a) with the associated uncertainty

( )

W

{

u

( )

E u

(

Rdig

)

u

(

RdigL

)

u

(

Rrep

)

}

u * = 2 + 2

δ

0 + 2

δ

+ 2

δ

(7.4-2a)

*

W and u

( )

W* can be determined directly using the information, and the results of the calibration as given in the calibration certificate:

Data sets Ical, Ecal, U

(

Ecal

)

, and/or

A characteristic E

( )

R = f

( )

I and U

(

E

( )

R

)

= g

( )

I . This is done in 7.4.1. and in 7.4.2.

To take account of the remaining possible influences on the weighing result, further corrections are formally added to the reading in a general manner resulting in the weighing result in general:

proc instr R

R W

W = *+δ +δ (7.4-1b)

with the associated uncertainty

( )

W u

( )

W u

(

Rinstr

)

u

(

Rproc

)

u = 2 * + 2

δ

+ 2

δ

(7.4-2b)

The added terms and the corresponding standard uncertainties are discussed in 7.4.3 and 7.4.4. The standard uncertainties u

( )

W* and u

( )

W are finally presented in 7.4.5.

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uncertainty of weighing results obtained under his normal conditions of use. They are not meant to be exhaustive nor mandatory.

Where a calibration laboratory offers such estimates to its clients which are based upon information that has not been measured by the laboratory, the estimates may not be presented as part of the calibration certificate.

7.4.1 Standard uncertainty of a reading in use

To account for sources of variability of the reading, (7.1.1-1) applies, with I

replaced by R:

(

dig

)

{

ecc

}

rep digL L R R R R R R R= +δ +δ − +δ L +δ 0 0 (7.4.1-1)

Term in

{ }

to be added if applicable The corrections and their standard uncertainties are:

7.4.1.1 δRdig0 accounts for the rounding error at zero reading. 7.1.1.1 applies with the exception that the variant dT <d, is excluded, so

(

R 0

)

d0 12

dig = (7.4.1-2)

Note 2 in 7.1.1.1 applies.

7.4.1.2 δRdigL accounts for the rounding error at load reading. 7.1.1.2 applies with the exception that the variant dT <dL is excluded, so

(

RdigL

)

dL 12

uδ = (7.4.1-3)

7.4.1.3 δRrep accounts for the error due to imperfect repeatability. 7.1.3.1 applies, the relevant standard deviation s or s

( )

I for a single reading, is to be taken from the calibration certificate, so

(

R

)

s

rep = or u

(

δRrep

)

=s

( )

R (7.4.1-4) Note: In the calibration certificate, the standard deviation may be reported as being related to a single indication, or to the mean of n indications. In the latter case, the value of s has to be multiplied by n to give the standard deviation for a single reading.

7.4.1.4 δRecc accounts for the error due to off-centre position of the centre of gravity of a load. It has been put in brackets as it is normally relevant only for W not for W*, and will therefore be considered in 7.4.4.3.

7.4.1.5 The standard uncertainty of the reading is then obtained by

( )

2 2 2

( )

{

2

(

)

2

}

0 2 ˆ 12 12 d s R w R R d R u = + R + L + ecc (7.4.1-5)

Term in

{ }

to be added if applicable

Note: the uncertainty u

( )

R is = constant where s = constant; where in exceptional cases the eccentricity error has to be considered, the term should be taken from 7.4.4.4.

(34)

7.4.2 Uncertainty of the error of a reading

Where a reading R corresponds to an indication Icalj reported in the calibration certificate, u

( )

Ecalj may be taken from there. For other readings, u

(

E

( )

R

)

may be calculated by (7.1.3-2) if

α

and β are known, or it results from interpolation, or from an approximation formula as per 7.2.

The uncertainty u

(

E

( )

R

)

is normally not smaller than u

( )

Ecalj for an indication Ij that is close to the actual reading R, unless it has been determined by an approximation formula.

Note: the calibration certificate normally presents U95

(

Ecal

)

from which u

(

Ecal

)

is to be derived considering the coverage factor k stated in the certificate.

7.4.3 Uncertainty from environmental influences

The correction term δRinstr accounts for up to 3 effects which are discussed hereafter. They do normally not apply to instruments which are adjusted right before they are actually being used – cf 4.2.4, case A. For other instruments they should be considered as applicable. No corrections are actually being applied, the corresponding uncertainties are estimated, based on the user’s knowledge of the properties of the instrument.

7.4.3.1 A term δRtemp accounts for a change in the characteristic (or adjustment) of the instrument caused by a change in ambient temperature. A limiting value can be estimated to δRtemp =TK∆T with the following terms.

Normally there is a manufacturer’s specification like TK =∂I

(

Max

)

∂T , in many cases quoted as TK ≤ TC in 10-6/K. By default, for instruments with type approval under OIML 76 [2], it may be assumed TC ≤mpe

(

Max

)

(

Max∆TAppr

)

where ∆TAppr is the temperature range of approval marked on the instrument; for other instruments, either a conservative assumption has to be made, leading to a multiple (3 to 10 times) of the comparable value for instruments with type approval, or no information can be given at all for a use of the instrument at other temperatures than that at calibration.

The range of variation of temperature ∆T (full width) should be estimated in view of the site where the instrument is being used, as discussed in Appendix A.2.2. Rectangular distribution is assumed, therefore the relative uncertainty is

(

)

12

ˆ R TC T

w temp = ∆ (7.4.3-1)

7.4.3.2 A term δRbouy accounts for a change in the adjustment of the instrument due to the variation of the air density; no correction to be applied, uncertainty contribution to be considered as in 7.1.2.2, where a variability of the air density larger than that at

Figure

Figure  4.2-1  shows  examples  for  the  magnitude  of  the  relative  changes
Figure 4.2-2 gives an impression of the magnitude of the apparent change in mass  in relation to a temperature difference, for some selected weight values
Fig. 5.3-1   Positions of load for test of eccentricity
Table 7.1-1 Quotient  c class = mpe m   for standard weights  m  ≥   100 g according to R 111 [3]
+7

References

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